Solving Dynamical Systems Involving Piecewise Restoring Force Using State Event Location

• Published in: Journal of engineering mechanics 2012-08, Vol.138 (8), p.997-1020
• Main Authors: Wright, Joseph P, Pei, Jin-Song
• Format: Article
• Language: English
• Subjects: Computer simulation; Dynamical systems; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Mathematical analysis; Mathematical models; Matlab; Nonlinear dynamics; Nonlinear dynamics and nonlinear dynamical systems; Physics; Robustness (mathematics); Runge-Kutta method; Solid mechanics; Statistical physics, thermodynamics, and nonlinear dynamical systems; Structural and continuum mechanics; Technical Papers
• ISSN: 0733-9399; 1943-7889
• DOI: 10.1061/(ASCE)EM.1943-7889.0000404

Abstract Many theoretical and experimental studies of complex path-dependent dynamic systems lead to restoring forces expressed as piecewise nonlinear algebraic equations. Examples include, but are not limited to, bilinear hysteretic, Ramberg-Osgood, Masing, generalized Masing, Clough, and Takeda models, which are popular in engineering mechanics applications. These models relate restoring force to displacement and velocity by means of piecewise relations having only C0 continuity, which leads to two sorts of challenges in numerical simulation. First, the equations of motion may not simply be a set of ordinary differential equations, rather they may fall within the framework of differential-algebraic equations (DAEs). Second, there are unknown locations of discontinuities of low-order derivatives of the solution. This study seeks accurate and efficient numerical solutions of the DAEs with C0 continuity, enabling robust simulation of these complex nonlinear dynamic systems. This study focuses on explicit time integration for single degree-of-freedom problems, while presenting a suitable problem formulation, detailed guidelines, case studies, and convincing insights, while exploiting two built-in MATLAB functions (ode45.m and the Events option). User-defined options are carefully examined, and recommendations are made based on a systematic study of approximation accuracy and computational efficiency, particularly as they relate to global error and tolerance proportionality when using an explicit, adaptive Runge-Kutta (RK) solver. Obtaining accurate values of state event locations results in a robust approach to solving the identified class of problems. This work initiates the possibility of treating many similar models by using the proposed programming module and, more importantly, by applying and further advancing the underlying theoretical concepts.